## Let the function (y = fleft( x right)) have the derivative (f’left( x right) = {left( {x – 2} right)^2}left( {1 – x} right)) for all ( x in mathbb{R}). The given function is covariate on which of the following intervals? – Math book

Given the function $$y = f\left( x \right)$$ have derivatives $$f’\left( x \right) = {\left( {x – 2} \right)^2}\left( {1 – x} \right)$$ with everyone $$x \in \mathbb{R}$$. The given function is covariate on which of the following intervals?

A. $$\left( {1;2} \right)$$.

B. $$\left( {1; + \infty } \right)$$.

C. $$\left( {2; + \infty } \right)$$.

D. $$\left( { – \infty ;1} \right)$$.

We have $$f’\left( x \right) > 0 \Leftrightarrow {\left( {x – 2} \right)^2}\left( {1 – x} \right) > 0 \Leftrightarrow \left\{ \ begin{array}{l}1 – x > 0\\{\left( {x – 2} \right)^2} > 0\end{array} \right \Leftrightarrow \left\{ \begin{array} {l}x < 1\\x \ne 2\end{array} \right. \Leftrightarrow x < 1$$.
So the function is covariant on the interval $$\left( { – \infty ;1} \right)$$.